Decoding FRB energetics and frequency features hidden by observational incompleteness

Fast radio bursts (FRBs) are millisecond-duration radio flashes of extragalactic origin, with magnetars implicated as viable central engines. Yet their triggering and radiation mechanisms remain unknown. Radio telescopes inevitably record bursts incompletely, as limited sensitivity and finite bandwidth lead to observational truncation. Here we establish a general analytical framework to reconstruct intrinsic population-level frequency characteristics and energetic parameters directly from observationally truncated FRB data. Applying this method to 2,223 bursts of FRB 20121102A observed by three different telescopes, we show that the narrow spectra of repeating FRBs are predominantly an observational selection effect. Only intrinsically high-energy bursts are genuinely narrowband. We further quantify, for the first time, the number and energy of completely undetected bursts, and reveal intrinsic long-term frequency evolution of the source. Our methodology transforms incomplete archival observations into physically meaningful probes, bridging instrumental readouts and intrinsic FRB physics.